MAST20009: Vector Calculus

4 Assignments (5% each), 1 Final exam (80%)

Lectures
Christine is an excellent lecture; very clear, precise and focused. Her explanations were great, and the subject coordination overall was fantastic. There was quite a bit of hand-holding though (e.g. demonstrating how to evaluate a determinant painstakingly each and every time), but you could just fast-forward through those bits. Clarity and precision is definitely to be preferred over vagueness.

Consulation
This year consultation was moved online via email. You'd send in an email with your questions and working at any time, and Christine would respond and provide guidance/answers as appropriate ^{quite quickly too!}. I found these too be really quite useful, and made frequent use of these to ask questions, or just regarding any nuance I came across that wasn't covered in the lectures themselves. I hope this get continued for other subjects next semester.

Tutorials
Tutorials were fantastic! I had Andrei Ratiu, who was amazing and super calm, and would always include extra challenges or teach interesting extended content if there was time at the end of the lesson. The only inconvenience was that it was difficult for those without a stylus/touch screen to be able to collaborate properly in the tutorial given the current circumstances.

Assignments
Honestly I found the assignments to be quite easy, scoring 100% on 3/4 of them. They are generally quite straight-forward applications of the content in the lectures, and often have direct parallels in the lecture notes. If you pay careful attention to the (checking orientation, notation, drawing graphs really carefully), it shouldn't be hard to score well.

Exam
The exams are quite straight forward, and don't seem to change much each year. There are generally no tricky question and no real proof questions (more just 'show that' procedural style questions), just be careful to not make any careless mistakes as usual . Also, pay attention to notation (e.g. Christine prefers a tilde under vector operators), there are often dedicated marks for correct use of notation, so don't just throw them away! If you have completed the problem book questions and past exams, then the exam shouldn't be any surprise.
The only reason I gave this subject 4.5 instead of 5 is that I felt there was much scope and content that could have been covered in greater depth and rigour than were (e.g. TNB frames, flowlines, conservative vector fields...). Vector Calculus just felt like an extension of Calc 2 (or UMEP), and wasn't really that difficult. Overall, it was an extremely enjoyable and well-taught maths subject.

Yes, Christine split lectures into smaller chunks and uploaded to LMS at beginning of week

Christine Manglesdorf

Yes, 5 past exams with brief answers/working

Nothing required (excepting lecture slides), Vector Calculus by Marsden and Tromba is recommended for supplementary reading, but I found Vector Calculus by Colley better.

3 lectures/week (~ 1 hour), 1 tutorial class (1 hour)

2020 Semester 1

- Four written assignments (5% each)
- 3-hour written exam (80%)

MAST20009 Vector Calculus is really the first subject that combines students coming from both the first year accelerated stream and the main stream, and is a must for students who wish to pursue applied maths, pure maths, physics or mathematical physics. The subject essentially takes what you learn in first year into higher dimensions.

The subject is broken into 6 sections.

Section 1: Functions of Several Variables
This section looks at limits, continuity and differentiability of functions of several variables (rather non-rigorously), as well as the chain rule for multiple variables. It also introduces the Jacboi matrix (derivative matrix) and the Jacobian for change of variables later, and also the matrix version of the chain rule. You will also look at Taylor polynomials for functions of several variables and error estimation. Locating critical points and extrema of functions will be revisited and you'll be introduced to Lagrange multipliers which are applied to optimisation problems with one or more constraints.

Section 2: Space Curves and Vector Fields
This section revises concepts from Specialist Maths / Calculus 1: parametric paths, and its properties such as velocity, speed and acceleration, as well as arc length. Concepts such as the unit tangent, unit normal, unit binormal, curvature and torsion are also seen, along with the Frenet-Serret frame of reference of a particle travelling on a path. In vector fields, you will look at ideas such as divergence, curl and Laplacian. Studied is also an informal look into flow lines of velocity fields and other useful things used later such as scalar and vector potentials.

Section 3: Double and Triple Integrals
This section is pretty self explanatory. Here you will learn how to evaluate double and triple integrals and put them to use against some physical problems (such as finding volumes, areas, masses of objects, centre of mass of an object, moment of inertia, etc). This section also discusses 3 important coordinate systems (polar, cylindrical and spherical) before finally diving into change of variables for multiple integrals.

Section 4: Integrals over Paths and Surfaces
Here, you will learn about path integrals, line integrals and surface integrals and apply them to some simple physical problems such as finding: total charge on a cable, mass of a rope, work done by a vector field on a particle, surface area of an object, flux, etc.

Section 5: Integral Theorems
The previous 4 sections build up to this. Finally, we have the required theory to understand the whole point of the subject. Here, you will use and apply Green's Theorem, the Divergence Theorem in the Plane, Stokes' Theorem (a basic version of it) and Gauss' Divergence Theorem, which make your life so much easier. You also get to apply some theory regarding scalar potentials and conservative vector fields studied in section 2. Some direct applications of the integral theorems include Gauss' Law and the continuity equation for fluid flow (latter not examinable). Those who are studying physics might want to look into Maxwell's equations for electromagnetic fields too (these are not examinable).

Section 6: General Curvilinear Coordinates
Here, we get to generalise some theory regarding coordinate systems. This makes our lives easier when dealing with a coordinate system that you may have not studied before, such as oblate spheroidal coordinates. Some connections to concepts back in sections 1 and 3 are also drawn.


So, what do I think of this subject?

Lectures
As a student who came from the accelerated stream, I can say this subject is markedly easier than AM2. From what I've heard from some mates, the pace of the subject is much like Calculus 2 and Linear Algebra. I personally felt that the subject was a bit slow. We spent so much time in lectures on just performing calculations rather than looking at the theory in any sort of depth. It got to the point where I didn't want to attend lectures because it just got so boring. (Like, yes, I think we know how to integrate \(\sin^2(x)\) with respect to \(x\). Other than that, Christine is a great lecturer and is very easy to understand.

Assignments
There are 4 of them and they are incredibly tedious. They're not hard at all. It's just a calculations fest. The questions basically consist of more tedious exam questions. Eg: here's a region, calculate its area. It's really not hard to full score the assignments. Just pull up Wolfram Alpha or use a CAS to check your calculations. Be careful to justify everything and be wary of direction and cheeky negative signs.

Tutorials
These are the best classes. You just get a sheet of problems and you complete them in small groups on the whiteboard while the tutors watch over your working and make any necessary corrections. I had a great tutor, and since I had my tutorial classes on Friday afternoons, it was a pretty small class and we had great banter. Nothing much else to say (other than "Will, you're a legend").

ExamLike most MAST subjects, it's 3 hours and worth 80%. There are an insane amount of past exams available. Do as many as you can. Doing well in this subject is about practice.

Yes, but only the main document camera.

Dr. Christine Mangelsdorf

Yes, lots and lots, some with answers.

4 out of 5

The lecture slides is available at the Co-op book shop. The lecture notes are online too, but I definitely recommend getting the book. It comes with a printed booklet of the problems sheets too. If you want more material to read, Vector Calculus by Marsden and Tromba (really any recent edition) is great, especially for those who prefer a higher level of mathematical rigor.

- Three 1-hour lectures
- One 1-hour tutorial

2019 Semester 1

96 (H1)

4 assignments throughout semester (5% each), end of semester exam (80%)

As I've seen in other reviews, this subject is commonly seen as the 'maths methods' of university maths at Melbourne, and rightly so. Whilst the concepts and mathematical tools introduced are a lot of fun (basically generalising all of single variable calculus to multiple variables and then some), I felt like I was having a whole bunch of formulas, methods and definitions stuffed down my throat, with minimal justification. And when there was 'justification', it was usually geometric/visual/intuitive and not rigorous.

This subject nicely relates to physics (a very high proportion of the students taking this subject are intending physics majors, presumably with the rest being mathematics majors), and many of the concepts introduced (such as line integrals, surface integrals) are related to physics concepts (work, flux). Hence, having some physics background (say VCE Physics) is nice to have, and makes lectures a little more lively and interesting. Then again, if you're doing this subject, you're likely doing some physics too.

In my case, I took this subject in my first semester of first year (having done UMEP Maths with my VCE). However, as I've heard, the UMEP course is different in 2016 which means students don't have to 'jump' AM2, so I won't say much about the gap (I think stolenclay's review does sufficient justice)

What was different this year from previous years was the lecturer - A/Prof. Andrei Ratiu, lecturing this subject for the first time. From the outset, Andrei was an excellent lecturer, and made the lectures worth attending, despite the course being at-times dry in terms of content. Andrei explained and demonstrated the concepts very well (albeit visually, but then I happen to be a proof pedant so don't mind me ), often using computer demonstrations to help us visualise things.

edit: I forgot to mention, Andrei's sense of humour is at times nothing short of charming

The prescribed materials comprise of partial lecture notes and a problem booklet. The partial lecture notes are quite essential for the lectures, as the lecturer usually doesn't do any writing - he just covers solutions with a piece of paper initially and gradually uncovers the solutions step-by-step with explanation. Students seem to have developed two approaches to doing the example problems that we went through in lectures:

• Copy the worked solutions from the document camera verbatim and not do any computation
• Not look or listen to the document camera/lecturer and work out the question on their own, then compare with the final answer on document camera

Usually, the latter is the more beneficial method. However, the former is suitable when you have become lost and just want to get down the solution for later study (as was the case when we did Taylor polynomials!)

The problems sheet comprehensively covers the types of questions which can be asked on the exam and on assignments, and I'd say doing all the questions is a must for every student. The vast majority (if not all) of the questions are not hard in the problem-solving sense, although they can get very computationally involved. For those who have a good grasp of the concepts, the main source of mistakes are simple algebra and arithmetic errors.

The assignments were very well set and fairly marked, and I believe that, putting in the requisite amount of time and attention, you can get a fairly good contribution to your mark without too much difficulty.

The exam itself was a bit of a wet blanket, definitely harder than 2015 and 2013 in my opinion. Conceptually, there was nothing difficult with the exam, most of the questions were the routine type. What was hard (and what consequently tripped me up on the exam) was that the questions were computationally difficult. Having been lulled into a false sense of security (by the 2015 sem 2 exam that most students (including myself) had left till the night before the exam), my anticipation of a similar 2016 exam were likely the cause of some below-expected performance on the exam.

Overall, I definitely did enjoy this subject, although I do lament the lack of proof and rigour (and have bad memories of the exam haha)

Yes, but the lecturer alternates between using two document cameras AND occasionally using the blackboard, so if you watch lecture recordings you'll only see 50% of the actual lecture.

A/Prof. Andrei Ratiu (such a nice lecturer)

Yes, 2013-2015, both semesters so a total of 6, with solutions.

4 Out of 5

Required: Partial lecture notes (only available in print) Recommended: Marsden and Tromba - Vector Calculus. To be honest, the lecture notes and prescribed exercises are more than sufficient to ensure coverage of the course content. As for the textbook, that's useful for proofs of theorems/formulae (sadly omitted at large from the subject), but you can definitely use any other vector calculus book such as Paul's Math Notes or other ones available free online (ask Google)

3x1 hour lectures per week, 1x1 hour tutorial per week

2016 Sem 1

H1 (91)

Four written assignments (totalling 20%); One 3 hour examination (80%)

This subject is actually quite boring. It is extremely important if you want to study physics like myself, or engineering, or if you want to study applied maths of any sort. But if you like pure maths I suspect you will hate this subject. It's also very easy - in many respects it's the VCE Methods of undergraduate mathematics.

Topics covered:

• Limits, continuity, differentiability, chain rule for partial derivatives, Taylor polynomials, Hessian & Jacobi matrices, Lagrange Multipliers
• Vector fields & streamlines, divergence, curl, laplacian, arclengths, tangents and normal vectors, curvature, torsion, scalar and vector potentials
• Double and triple integrals in Cartesian, Polar, Cylindrical and Spherical co-ordinates
• Line integrals over scalar and vector fields, parametrisation of surfaces, surface integrals of scalar functions, surface integrals over vector fields (i.e. flux integrals)
• Green's theorem, Stokes' Theorem, Gradient Theorem for conservative fields, Divergence Theorem
• General curvilinear coordinates

The issue is that 6. is vitally important in physics applications such as General Relativity but is (as usual) rushed at the end. 1. and 2. are very easy, as is 3. The only challenging aspects of the course, in my opinion, are surface (and flux) integrals along with the various integral theorems. Note, however, that in physics and engineering that these are probably the most important sections of the course.

I do recommend getting a hold of the completed notes as early as you can and working through them by yourself, which leaves you with enough time to revise. The assignments are quite easy, and the exam was tougher but manageable overall. Frankly, if you are not overconfident ahead of the exam as I was, this is your best chance in second-year mathematics, physics or perhaps engineering to improve your GPA with a mark of >90%.

Yes, with screen capture.

Dr. Mark Fackrell. He looks a little like a long-haired William H. Macy, but he's a decent lecturer.

Yes, there were plenty.

4 Out of 5

None. The 'partial' lecture notes are available at the Co-op Bookshop. In Semester 1, 2013 (but not Semester 2) they were also available on the LMS. Note, however, that these were in fact full (and not partial) lecture notes, with the 'completed' worked examples in white - just use Adobe Acrobat etc. to render this black. Alternatively, ask to join the 2013 Semester 1 Facebook group for the subject where the preformatted full lecture notes have been posted.

3 x 1 hour lectures per week; 1 x 1 hour tutorial per week

Semester 1, 2013

82